Discrete scale invariance, complex fractal dimensions, and log‐periodic fluctuations in seismicity

Abstract

We discuss in detail the concept of discrete scale invariance and show how it leads to complex critical exponents and hence to the log‐periodic corrections to scaling exhibited by various measures of seismic activity close to a large earthquake singularity. Discrete scale invariance is first illustrated on a geometrical fractal, the Sierpinsky gasket, which is shown to be fully described by a complex fractal dimension whose imaginary part is a simple function (inverse of the logarithm) of the discrete scaling factor. Then, a set of simple physical systems (spins and percolation) on hierarchical lattices is analyzed to exemplify the origin of the different terms in the discrete renormalization group formalism introduced to tackle this problem. As a more specific example of rupture relevant for earthquakes, we propose a solution of the hierarchical time‐dependent fiber bundle of Newman et al. [1994] which exhibits explicitly a discrete renormalization group from which log‐periodic corrections follow. We end by pointing out that discrete scale invariance does not necessarily require an underlying geometrical hierarchical structure. A hierarchy may appear “spontaneously” from the physics and/or the dynamics in a Euclidean (nonhierarchical) heterogeneous system. We briefly discuss a simple dynamical model of such mechanism, in terms of a random walk (or diffusion) of the seismic energy in a random heterogeneous system.